A study on convergence of non-convolution type double ...

NTMSCI 4, No. 4, 67-78 (2016) 67

New Trends in Mathematical Scienceshttp://dx.doi.org/10.20852/ntmsci.2016422309

A study on convergence of non-convolution type doublesingular integral operatorsMine Menekse Yilmaz

Department of Mathematics, Faculty of Arts and Sciences, Gaziantep University, Gaziantep, Turkey

Received: 18 May 2016, Accepted: 8 September 2016Published online: 16 October 2016.

Abstract: The aim of this paper is to investigate the pointwise convergence and the rate of convergence of the operators in the followingform:

Lλ ( f ;x,y) =∫∫

Ω

f (t,s)Kλ (t,s;x,y)dsdt, (x,y) ∈ Ω ,

whereΩ =< A,B > × < C,D > is an arbitrary closed, semi-closed or open region inR2, at a µ-generalized Lebesgue point of

f ∈ Lp(Ω) as(x,y,λ )→ (x0,y0,λ0) .

Keywords: µ-generalized Lebesgue points, pointwise convergence, rate of convergence.

1 Introduction

In [11], some pointwise approximation results for integral operators of the form:

Uλ ( f ;x) =

π∫

−π

f (t)Kλ (t − x)dt, x∈ (−π ,π), (1)

have been studied at a Lebesgue point of integrable functions. Then these type operators, convolution type singularintegral operators depending on two parameters, were developed by Gadjiev [3], Rydzewska [7]. In [12], Taberski gavesome theorems concerning the Riemann-Stieltjes, Lebesgueand Titchmarsh integrals on a given rectangle. Especially,heformulated a theorem of Faddeev’s type ( [12], page 246) concerning the convergence of singular integrals of the form:

U (x,y;λ , f ) =∫∫

Q

f (t,s)ψ (x,y;λ )dsdt, (x,y) ∈ Q, (2)

for integrable functionsf , here Q denotes a given rectangle andψ (x,y;λ ) is the kernel satisfying suitable assumptions.In the same paper, he explored the some approximation theorems by using the following operators:

V (x,y;λ , f ) =

π∫

−π

π∫

−π

f (t,s)K (t − x,s− y;λ )dsdt (3)

In the summability theory of double Fourier series the integral operators of type (3) play an important role, where kernelK (t,s;λ ) are bounded, measurable, even, 2π periodic in each variablex, y separately. After the pioneering work of

∗ Corresponding author e-mail:[email protected] c© 2016 BISKA Bilisim Technology

http://dx.doi.org/10.20852/ntmsci.2016422309

68 M. Menekse Yilmaz: A study on convergence of non-convolution type double singular integral operators

Taberski [12], in [8], Rydzewska estimated the order of convergence of double singular integrals of the form

V (x,y;σ , f ) =

π∫

−π

π∫

−π

f (t,s)K (t − x,s− y;σ)dsdt (4)

under various assumptions onf (s, t) andK (t,s;σ) . Rydzewska considered the convergence of integral operators to areal integrable functionf (t,s) at a generalized Lebesgue point. In particular, for furtherstudies on the convergence ofdouble singular integrals at the Lebesgue points, we address the reader to [9], [10], [14],[15] and [16]. For further readingwe suggest the following papers : [1], [5], [17], [18], [19], and [21]-[24]. We present the pointwise convergence of non-convolution singular integral operators of the form

Lλ ( f ;x,y) =∫∫

Ω

f (t,s)Kλ (t,s;x,y)dsdt, (x,y) ∈ Ω , (5)

whereΩ := 〈A,B〉× 〈C,D〉 is an arbitrary closed, semi-closed or open region inR2, at aµ−generalized Lebesgue point

of f ∈ Lp (Ω) as(x,y,λ ) → (x0,y0,λ0) . HereLp (Ω) is the collection of all measurable functionsf for which | f |p isintegrable onΩ .

The rest of the paper is organized as follows. Section 2 introduces the terminology used throughout this paper. Section 3shows the existence of the operators of type (5). In Section 4, we give two theorems concerning the pointwiseconvergence ofLλ ( f ;x,y) on different regions. In Section 5 we estimate the rate of pointwise convergence of theoperators of type (5) and give some examples.

2 Preliminaries

Definition 1. Let ϕ (x,y) be a function defined in the rectangle D, let

P=

a= x1 < x2 < ... < xi < ... < xm < xm+1 = bc= y1 < y2 < ... < y j < ... < yn < yn+1 = d

be a partition of D and

ϕ (xi ,y j) = ϕ (xi ,y j)−ϕ (xi+1,y j)−ϕ(xi ,y j+1

)+ϕ

(xi+1,y j+1

).

If ϕ (xi ,y j) ≥ 0 (i = 1,2, ...,m; j = 1,2, ...n) for any partition of P, then it is said thatϕ (x,y) satisfies the conditionΩin D [12].

In other words, ifϕ (xi ,y j) ≥ 0 for all partitions of D then it is said thatϕ (x,y) is bimonotonically increasing and ifϕ (xi ,y j)≤ 0 for all partitions of D, then it is said thatϕ (x,y) is bimonotonically decreasing [4].

The definition of theµ−generalized Lebesgue point is the special form of the definition of the Lebesgue point in [8].

Definition 2. A point(x0,y0) ∈ D is called aµ−generalized Lebesgue point of function f∈ Lp (D) if

lim(h,k)→(0,0)

1µ1(h)µ2(k)

h∫

0

k∫

0

| f (t + x0,s+ y0)− f (x0,y0)|pdsdt= 0

whereµ1(t):R→ R, absolutely continuous on[−δ0,δ0] , increasing on[0,δ0] andµ1(0) = 0 and alsoµ2(s):R→ R, absolutely continuous on[−δ0,δ0] , increasing on[0,δ0] andµ2(0) = 0. Here,0< h,k< δ0.

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NTMSCI 4, No. 4, 67-78 (2016) /www.ntmsci.com 69

The following definition was inspired by the definition of class A in [13].

Definition 3. (Class A) Let Ω =< A,B> × <C,D >, Ψ =< a,b> × < c,d >, Λ ⊂ R+0 be an index set andλ ∈ Λ be

an accumulation point of it. If the following conditions aresatisfied, then Kλ : R2×R2 →R

+0 belongs to class A; i.e.,

(a) For fixed(x0,y0) ∈ Ω , Kλ (x0,y0) tends to infinity asλ tends toλ0 for any fixed(x,y) ∈< a,b>×< c,d > .

(b) lim(x,y,λ )→(x0,y0,λ0)

∫∫

R2Kλ (t,s;x,y)dsdt= 1.

(c) For any fixed(x,y) ∈Ψ there exists a point(x0,y0) ∈ Ω such that

limλ→λ0

∫∫

R2\N

Kλ (t,s;x,y)dsdt= 0, ∀N ∈ N(x0,y0),

whereN(x0,y0) stands for the family of all neighborhoods of(x0,y0) in R2.

(d) For any fixed(x,y) ∈Ψ there exists a point(x0,y0) ∈ Ω such that

limλ→λ0

supR2\Nδ (x0,y0)

Kλ (t,s;x,y) = 0, ∀δ > 0

whereNδ (x0,y0) = (x0− δ ,x0+ δ )× (y0− δ ,y0+ δ ).(e) For any fixed x∈< a,b > there exists a point x0 ∈< A,B > such that Kλ (t,s;x,y) is monotonically increasing

with respect to t on< x0 − δ ,x0 > and monotonically decreasing on< x0,x0 + δ > and similarly, for any fixedy0 ∈< C,D > there exists a point y∈< c,d >such that Kλ (t,s;x,y) is monotonically increasing with respect tos on< y0 − δ ,y0 > and monotonically decreasing on< y0,y0 + δ > for any λ ∈ Λ . Analogously, for any fixed(x,y) ∈Ψ there exists a point(x0,y0) ∈ Ω such that Kλ (t,s;x,y) is bimonotonically increasing with respect to(t,s)on< x0,x0+ δ > × < y0,y0+ δ > and< y0− δ ,y0 > × < x0− δ ,x0 > and similarly bimonotonically decreasingwith respect to(t,s) on< x0,x0+ δ >×< y0− δ ,y0 > and< x0− δ ,x0 >×< y0,y0+ δ > for anyλ ∈ Λ .

Throughout this paper, we suppose that the kernelKλ (t,s;x,y) belongs to classA.

3 Existence of operator

Lemma 1. Let ‖Kλ (., .;x,y)‖L1(R2) ≤ M, ∀λ ∈ Λ and ∀(x,y) ∈ Ψ . If f ∈ L1(Ω) then Lλ ( f ;x,y) defines a continuoustransformation acting on L1(Ω).

Proof.By the linearity of the operatorLλ ( f ;x,y) , it is sufficient to show that

‖Lλ‖1 = supf 6=0

‖Lλ ( f ,x,y)‖L1(Ω)

‖ f‖L1(Ω)

< ∞

remains bounded. Now, using Fubini Theorem [2] we can write

‖Lλ ( f ,x,y)‖L1(Ω) =

∫∫

Ω

∫∫

Ω

f (t,s)Kλ (t,s;x,y)dsdt

dydx

≤

∫∫

Ω

f (t,s)

∫∫

R2

Kλ (t,s;x,y)dydx

dsdt

≤ M‖ f‖L1(Ω) .

Thus the proof is completed.


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Lemma 2.Let 1< p< ∞ and‖Kλ‖Lq(R2×R2) ≤ M, ∀λ ∈ Λ whenever1p +1q = 1. If f ∈ Lp(Ω) then Lλ ( f ;x,y) defines a

continuous transformation from Lp(Ω) to Lq(Ω).

Proof.We assume that 1< p< ∞. By the linearity of the operatorLλ ( f ;x,y) , it is sufficient to show that

‖Lλ‖q = supf 6=0

‖Lλ ( f ,x,y)‖Lq(Ω)

‖ f‖Lp(Ω)

is bounded. Let us define a new function by

g(t,s) =

f (t,s) ,(t,s) ∈ Ω ,

0,(t,s) ∈ R2\Ω .

Rearranging and rewriting the norm as follows

‖Lλ ( f ,x,y)‖Lq(Ω) = ‖Lλ (g,x,y)‖Lq(Ω) =

∫∫

Ω

∣∣∣∣∣∣

∫∫

R2

g(t,s)Kλ (t,s;x,y)dsdt

∣∣∣∣∣∣

q

dydx

1q

.

Applying Holder’s inequality [2] to the last equality we obtain

‖Lλ ( f ,x,y)‖Lq(Ω) ≤

∫∫

Ω

∫∫

R2

|g(t,s)|pdsdt

1p∫∫

R2

|Kλ (t,s;x,y)|qdsdt

1q

q

dydx

1q

≤

∫∫

Ω

‖ f‖qLp(Ω)

∫∫

R2

|Kλ (t,s;x,y)|qdsdt

dydx

1q

≤ ‖ f‖Lp(Ω) ‖Kλ‖Lq(R2×R2)

≤ M ‖ f‖Lp(Ω) .

Hence the proof is completed.

4 Convergence at characteristic points

We are now ready to prove our first main result.

Theorem 1.If (x0,y0) be a generalized Lebesgue point of function f∈ Lp (Ω) then

lim(x,y,λ )→(x0,y0,λ0)

Lλ ( f ;x,y) = f (x0,y0)

on any set Z on which the function

x0+δ∫

x0−δ

y0+δ∫

y0−δ

Kλ (t,s;x,y)∣∣µ1 (|t − x0|)

′t

∣∣ ∣∣µ2 (|s− y0|)′s

∣∣dsdt

is bounded as(x,y,λ ) tends to(x0,y0,λ0).

Proof. Suppose thatNδ (x0,y0) ⊂ Ω and(x0,y0) be µ−generalized Lebesgue point off ∈ Lp (Ω) . For the casep = 1,the proof is quite similar to forp > 1, therefore we will prove the theorem for the case 1< p < ∞. Since(x0,y0) ∈ Ω



be µ−generalized Lebesgue point off ∈ Lp (Ω) , givenε > 0, there exists aδ > 0 such that for allh andk satisfying0< h,k≤ δ , the following inequalities

x0∫

x0−δ

y0∫

y0−δ

| f (t,s)− f (x0,y0)|pdsdt< εµ1 (h)µ2 (k) , (1)

x0∫

x0

+δy0∫

y0−δ

| f (t,s)− f (x0,y0)|pdtdts< εµ1 (h)µ2 (k) , (2)

x0∫

x0−δ

y0+δ∫

y0


x0+δ∫

x0

y0+δ∫

y0


hold. SetIλ (x,y) := |Lλ ( f ;x,y)− f (x0,y0)|. According to condition (c) of classA, we shall write

∣∣∣∣∣∣

∫∫

Ω

f (t,s)Kλ (t,s;x,y)dsdt− f (x0,y0)

∣∣∣∣∣∣

≤

∫∫

Ω

| f (t,s)− f (x0,y0)|Kλ (t,s;x,y)dsdt+ | f (x0,y0)|

∣∣∣∣∣∣

∫∫

R2

Kλ (t,s;x,y)dsdt−1

∣∣∣∣∣∣

+ | f (x0,y0)|∫∫

R2\Ω

Kλ (t,s;x,y)dsdt

= I + II + III .

By condition (c) of classA, III → 0 asλ → λ0 respectively. Now, we investigate the integralI + II . Using Holder’sinequality we have the following

I + II ≤

∫∫

Ω

| f (t,s)− f (x0,y0)|p |Kλ (t,s;x,y)|dsdt

1p

×

∫∫

Ω

Kλ (t,s;x,y)dsdt

1q

+ | f (x0,y0)|

∣∣∣∣∣∣

∫∫

R2


∣∣∣∣∣∣.

Since wheneverm,n positive numbers the inequality(m+n)p ≤ 2p(mp+np) holds [6], by taking thep− th power of bothsides we have

(I + II )p ≤ 2p∫∫

Ω


×

∫∫

Ω

Kλ (t,s;x,y)dsdt

pq

+2p | f (x0,y0)|p

∣∣∣∣∣∣

∫∫

R2


∣∣∣∣∣∣

p

= 2p(I1× I∗+ I2).


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Observe that by condition (c) of classA, the integralI∗ tends to 1 as(x,y,λ ) tends to(x0,y0,λ0) and the integralI2 tendsto zero. Now we investigate the integralI1.

I1 =

∫ ∫

Ω\Nδ (x0,y0)

+

∫ ∫

Nδ (x0,y0)

| f (t,s)− f (x0,y0)|

pKλ (t,s;x,y)dsdt= I11+ I12.

The following inequality holds for the integralI11 i.e.:

I11 ≤ supΩ\Nδ (x0,y0)

Kλ (t,s;x,y)2p[‖ f‖p

L1(Ω)+ | f (x0,y0)|

p |B−A| |D−C|].

Hence by condition (d) of classA, I11 → 0 as(x,y,λ )→ (x0,y0,λ0) .

Next, we can show thatI12 tends to zero as(x,y,λ )→ (x0,y0,λ0) on Nδ (x0,y0).

I12=

∫ ∫

Nδ (x0,y0)

| f (t,s)− f (x0,y0)|pKλ (t,s;x,y)dsdt

=

x0+δ∫

x0−δ

y0+δ∫

y0−δ


=

x0∫

x0−δ

y0∫

y0−δ

+

x0+δ∫

x0

y0∫

y0−δ

| f (t,s)− f (x0,y0)|

pKλ (t,s;x,y)dsdt

+

x0∫

x0−δ

y0+δ∫

y0

+

x0+δ∫

x0

y0+δ∫

y0


= I121+ I122+ I123+ I124.

Hence we can evaluate the integralI121. From [12] (see 2.5 p.101), we can write the following:

I121=

x0∫

x0−δ

y0∫

y0−δ

| f (t,s)− f (x0,y0)|pKλ (t,s;x,y)dsdt= (S)

x0∫

x0−δ

y0∫

y0−δ

Kλ (t,s;x,y)dF (t,s)

where(S) denotes Stieltjes integral.

Two-dimensional integration by parts (see 2.2 p.100 in [12]) give us

x0∫

x0−δ

y0∫

y0−δ

Kλ (t,s;x,y)dF (t,s) =

x0∫

x0−δ

y0∫

y0−δ

F (t,s)dKλ (t,s;x,y)

+

x0∫

x0−δ

F (t,y0− δ )dKλ (t,y0− δ ;x,y)+

y0∫

y0−δ

F (x0− δ ,s)dKλ (x0− δ ,s;x,y)

+F (x0− δ ,y0− δ )Kλ (x0− δ ,y0− δ ;x,y) .



From(1), we can write

I121≤ εx0∫

x0−δ

y0∫

y0−δ

µ1 (x0− t)µ2 (y0− s) |dKλ (t,s;x,y)|

+ εµ2(δ )x0∫

x0−δ

µ1 (x0− t) |dtKλ (t,y0− δ ;x,y)|

+ εµ1(δ )y0∫

y0−δ

µ2 (y0− s) |dsKλ (x0− δ ,s;x,y)|

+ εµ1(δ )µ2 (δ )Kλ (x0− δ ,y0− δ ;x,y)

= i1+ i2+ i3+ i4.

After applying integration by parts toi1, i2 andi3 we obtain the following result

I121≤ εx0∫

x0−δ

y0∫

y0−δ

Kλ (t,s;x,y)∣∣∣µ1 (x0− t)

′

t

∣∣∣∣∣∣µ2 (y0− t)

′

s

∣∣∣dsdt.

For the integralsI122, I123 andI124 the proof is similar to the above one. Thus we obtain the following inequalities:

I122≤−εx0+δ∫

x0

y0∫

y0−δ

Kλ (t,s;x,y)∣∣∣µ1 (t − x0)

′

t

∣∣∣∣∣∣µ2 (y0− s)

′

s

∣∣∣dsdt,

I123≤−εx0∫

x0−δ

y0+δ∫

y0

Kλ (t,s;x,y)∣∣∣µ1 (x0− t)

′

t

∣∣∣∣∣∣µ2 (y0− s)

′

s

∣∣∣dsdt,

I124≤ εx0+δ∫

x0

y0+δ∫

y0

Kλ (t,s;x,y)∣∣∣µ1 (t − x0)

′

t

∣∣∣∣∣∣µ2 (s− y0)

′

s

∣∣∣dsdt.

Collecting the estimatesI121, I122, I123 andI124 , we have the following inequality:

I12 ≤ I121+ I122+ I123+ I124= εx0+δ∫

x0−δ

y0+δ∫

y0−δ

Kλ (t,s;x,y)∣∣µ1 (|t − x0|)

′t

∣∣ ∣∣µ2 (|s− y0|)′s

∣∣dsdt.

Therefore, if the points(x,y,λ ) ∈ Z are sufficiently near to(x0,y0,λ0) , we have

I12 ≤ εK

where

K = sup

x0+δ∫

x0−δ

y0+δ∫

y0−δ

Kλ (t,s;x,y)∣∣µ1 (|t − x0|)

′t

∣∣ ∣∣µ2 (|s− y0|)′s

∣∣dsdt. : (x,y,λ ) ∈ Z

.

Thus, the proof is finished.

The following theorem gives a pointwise approximation of the integral operators type (3) to the functionf atµ-generalizedLebesgue point off ∈ L1(R

2) wheneverD = R2.


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Theorem 2.Suppose that the hypothesis of Theorem 1 is satisfied forΩ = R2. If (x0,y0) is a µ-generalized Lebesgue

point of function f∈ L1(R2) then

lim(x,y,λ )→(x0,y0,λ0)

Lλ ( f ;x,y) = f (x0,y0) .

Proof.Using the same strategy as in Theorem 1 we obtain

Iλ (x,y) := |Lλ ( f ;x,y)− f (x0,y0)|

≤

∫∫

R2

| f (t,s)− f (x0,y0)| |Kλ (t,s;x,y)|dsdt+ | f (x0,y0)|

∣∣∣∣∣∣

∫∫

R2


∣∣∣∣∣∣

= I + II

(I + II )p ≤ 2p∫∫

R2


×

∫∫

R2

Kλ (t,s;x,y)dsdt

pq

+2p | f (x0,y0)|p

∣∣∣∣∣∣

∫∫

R2


∣∣∣∣∣∣

p

= 2p(I1× I∗+ I2)

Observe that by condition (b) of classA, the integralI∗ tends to 1 as(x,y,λ )→ (x0,y0,λ0) .

I1 =

∫∫

R2\Nδ (x0,y0)

+

∫∫

Nδ (x0,y0)

| f (t,s)− f (x0,y0)|

p |Kλ (t,s;x,y)|dsdt

= I11+ I12.

I11 =∫∫

R2\Nδ (x0,y0)


≤∫∫

R2\Nδ (x0,y0)

2p| f (t,s)|p+ f |(x0,y0)|p|Kλ (t,s;x,y)|dsdt

=∫∫

R2\Nδ (x0,y0)

2p | f (t,s)|pKλ (t,s;x,y)dsdt+2p | f (x0,y0)|p

∫∫

R2\Nδ (x0,y0)

Kλ (t,s;x,y)dsdt

≤ supR2\Nδ (x0,y0)

Kλ (t,s;x,y)2p‖ f‖pL1(R2)

+2p | f (x0,y0)|p

∫∫

R2\Nδ (x0,y0)

Kλ (t,s;x,y)dsdt

Hence by condition (d) and (c) of class A,I11→ 0 asλ → λ0. Using the same operations for the integralI12 as in Theorem1, we have the following inequality:

I12 =∫∫

Nδ (x0,y0)


≤ εx0+δ∫

x0−δ

y0+δ∫

y0−δ

|Kλ (t,s;x,y)|∣∣∣µ1 (|t − x0|)

′

t

∣∣∣∣∣∣µ2 (|s− y0|)

′

s

∣∣∣dsdt.



Therefore, if the points(x,y,λ ) ∈ Z are sufficiently near to(x0,y0,λ0), we have

I12 ≤ εK

where

K = sup

x0+δ∫

x0−δ

y0+δ∫

y0−δ

|Kλ (t,s;x,y)|∣∣∣µ1 (|t − x0|)

′

t

∣∣∣∣∣∣µ2 (|s− y0|)

′

s

∣∣∣dsdt: (x,y,λ ) ∈ Z

.

Thus, the proof is finished.

5 Rate of convergence

In this section, we give a theorem concerning the rate of pointwise convergence.

Theorem 3.Suppose that the hypothesis of Theorem 1 is satisfied. Let

∆(λ ,δ ,x,y) =x0+δ∫

x0−δ

y0+δ∫

y0−δ

|Kλ (t,s;x,y)|∣∣∣µ1(|t − x0|)

′

t

∣∣∣∣∣∣µ2(|s− y0|)

′

s

∣∣∣dsdt

for 0< δ < δ0 and the following assumptions are satisfied:

(i) ∆(λ ,δ ,x,y)→ 0 as(x,y,λ )→ (x0,y0,λ0) for someδ > 0.(ii) For any fixed(x,y) ∈Ψ there exists a point(x0,y0) ∈ Ω such that

∫∫

R2\N

Kλ (t,s,x,y)dsdt= o(∆(λ ,δ ,x,y)),∀N ∈ N(x0,y0)

asλ −→ λ0.

(iii) For any fixed(x,y) ∈Ψ there exists a point(x0,y0) ∈ Ω such that

supR2\N

Kλ (t,s;x,y) = o(∆(λ ,δ ,x,y)) ,δ > 0,

asλ → λ0. Then at eachµ-generalized Lebesgue point of f∈ L1(Ω) and we have as(x,y,λ )→ (x0,y0,λ0)

|Lλ ( f ;x,y)− f (x0,y0)|p = o(∆(λ ,δ ,x,y)).

Proof.The result is clear by Theorem 1 and Theorem 2.

For the one-dimensional counterpart of the following kernel, we refer the reader to see [13].

Example 1.We defineKλ (t,s;x,y) =

λ 2

|xy| , i f (t,s) ∈ [0 xλ ]× [0, y

λ ], xy 6= 0,

0, i f (t,s) ∈ R2\[0, x

λ ]× [0, yλ ]

and consider the function

f (t,s) =1

(1+ t2)(1+ s2),(t,s) ∈ R

2.

Now first, we computeLλ and then we have the following:

Lλ ( f ;x,y) =∫∫

[0, xλ ]×[0,

yλ ]

1(1+ t2)(1+ s2)

λ 2

|xy|dsdt=

λ 2

|xy|arctan

(yx

)arctan

(yx

), xy 6= 0.

We give the graph off (t,s) = 1(1+t2)(1+s2)

, (t,s) ∈ R2 as shown in Figure 1:


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Fig. 1: Graph of f (t,s) = 1(1+t2)(1+s2)

Now we give the two graph ofLλ ( f ;x,y) as(x,y,λ ) → (0,0,∞). In the first graph we chooseλ = 10 in the second oneλ = 100.

Graph ofLλ ( f ;x,y) = λ 2

|xy| arctan( xλ )arctan( y

λ ), xy 6= 0 whenλ = 10, x∈ (0.01,1) andy∈ (0.01,1).

Fig. 2: Graph ofLλ ( f ;x,y) whenλ = 10

Graph ofLλ ( f ;x,y) = λ 2

|xy| arctan( xλ )arctan( y

λ ), xy 6= 0 whenλ = 100, x∈ (0.01,1) andy∈ (0.01,1)



Fig. 3: Graph ofLλ ( f ;x,y) whenλ = 100

Figures are generated by Wolfram Mathematica 7, we refer thereader to see [20].

References

[1] C. Bardaro and G. Vinti, On Approximation Properties of Certain Non-convolution Integral Operators, Journal of ApproximationTheory 62(1990), 358-371.

[2] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, vol. I. Academic Press, New York, London, (1971).[3] A.D. Gadjiev, On the order of convergence of singular integrals which depend on two parameters, Special Problems of Functional

Analysis and their Appl. to the Theory of Diff. Eq. and the Theory of Func., Izdat. Akad. Nauk Azerbaıdazan. SSR. (1968), 40–44.[4] S.R. Ghorpade, B.V. Limaye, A Course in Multivariable Calculus and Analysis, Springer, New York, (2010).[5] R. G. Mamedov, On the order of convergence of m-singular integrals at generalized Lebesgue points and in the space Lp(-∞, ∞),

Izv. Akad. Nauk SSSR Ser. Mat 27(1963), no.2, 287-304.[6] W. Rudin, Real and Complex Analysis, Mac Graw Hill Book Company, London, (1987).[7] B. Rydzewska, Approximation des fonctions par des integrales singulieres ordinaires, Fasc. Math. (7) (1973), 71–81.[8] B. Rydzewska, Approximation des fonctions de deux variables par des integrales singulieres doubles, Fasc. Math.(8) (1974),

35–45.[9] S. Siudut, On the convergence of double singular integrals, Comment. Math. Prace Mat. 28 (1) (1988), 143-146.

[10] S. Siudut, A theorem of Romanovski type for double singular integrals, Comment. Math. Prace Mat. 29 (1989), 277-289.[11] R. Taberski, Singular integrals depending on two parameters, Rocznicki Polskiego towarzystwa matematycznego, Seria I. Prace

matematyczne, VII, (1962), 173-179.[12] R. Taberski, On double integrals and Fourier Series, Ann. Polon. Math. 15 (1964), 97–115[13] H. Karsli, On Approximation Properties of Non-Convolution Type Nonlinear Integral Operator, Anal. Theory Appl.,Vol. 26, No.

2 (2010), 140-152[14] G. Uysal, M. M. Yilmaz, E. Ibikli, A study on pointwise approximation by double singular integral operators, J. Inequal. Appl.,

2015:94, 2015.[15] G. Uysal and M. M. Yilmaz, Some theorems on the approximation of non-integrable functions via singular integral operators,

Proc. Jangjeon Math. Soc., 18(2015), no. 2, 241-251.[16] G. Uysal and E. Ibikli, Further Results On Approximation By Double Singular Integral Operators with Radial Kernels, Journal of

Pure and Applied Mathematics: Advances and App.,Volume 14,Number 2, 2015, Pages 151-166.[17] G. Uysal , M. Menekse Yilmaz and E. Ibikli, Approximation by Radial Type Multidimensional Singular Integral

Operators,Palestine Journal of Mathematics, 2016, no:5(2), 61-70.


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[18] G. Uysal, E. Ibikli, Weighted approximation by double singular integral operators with radially defined kernels, Math. Sci.(Springer) 10 (2016), 1-9.

[19] M. M. Yilmaz, G. Uysal, and E. Ibikli, A note on rate of convergence of double singular integral operators, Adv. Difference Equ.,2014:287, 2014.

[20] Wolfram, S., The Mathematica Book, Fifth Edition (2003), Wolfram Media, Inc., 1488 pp.[21] L. N. Mishra, V. N. Mishra, K. Khatri, Deepmala, On The Trigonometric approximation of signals belonging to generalized

weighted Lipschitz W(L′r ,ξ (t))(r ≥ 1) class by matrix(C1.Np

)operator of conjugate series of its Fourier series, Applied

Mathematics and Computation, vol.237(2014) , 252-263.[22] V. N. Mishra, K. Khatri, L.N. Mishra, Deepmala, Trigonometric approximation of periodic signals belonging to generalized

weighted LipschitzW′(Lr ,ξ (t))(r ≥ 1) class by Norlund-Euler(N ,pn) belonging to operator of conjugate series of its Fourier

series, Journal of Classical Analysis, Vol.5, Number 2(2014), 91-95. doi:10.7153/jca-05-08.[23] V. N. Mishra, Some Problems on Approximations of Functions in Banach Spaces, Ph.D. Thesis (2007), Indian Instituteof

Technology, Roorkee 247 667, Uttarakhand, India.[24] V. N. Mishra, L.N. Mishra, Trigonometric Approximation of Signals (Functions) in Lp(p≥ 1) norm, International Journal of

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